dbart06 wrote:for all positive integers, [M]=3m when m is odd, [M]=1/2m when m is even, which is equivalent to [9]*[6].

a)81
b)54
c)37
d)27
e)18

sorry for the confusion.

That's not quite what the question says, if I remember this correctly from GMATPrep. The answer choices should be:

a) [81]
b) [54]
c) [37]
d) [27]
e) [18]

We know [m]=3m when m is odd, [m]=(1/2)*m when m is even. Thus:
[9] = 3*9 = 27
[6] = 6/2 = 3

and [9]*[6] = 27*3 = 81. This often leads people to choose answer choice A, but notice that answer choice A is not equal to 81. The [] function is applied to each answer as well; answer choice A is [81] = 3*81 = 243. D is the correct answer, because [27] = 3*27 = 81 = [9]*[6].

pre-gmat wrote:I think the problem can be solved much easily.

[9]*[6] is essentially [54] and this is an even number so as per the question M=1/2 m when m is even.

1/2 *[54] =27

Let me know if my reasoning is wrong.

You cannot multiply through functions in the way you're attempting to do- that is, f(x)*f(y) is not normally equal to f(x*y). [9]*[6] is not equal to [54], and is not equal to 27; as described above, it is equal to [27], or 81.